PORTUGALIAE MATHEMATICA
Vol. 57 Fasc. 3 – 2000
EXTENDED COMPOSITION OPERATORS
IN WEIGHTED SPACES
L. Oubbi
Abstract: Let X and Y be Hausdorff completely regular spaces and βX the StoneČech compactification of X. For locally convex spaces E and F consisting of continuous
functions respectively on X and Y and whose topologies are generated by seminorms
that are weighted analogues of the suprimum norm, we give necessary and sufficient
conditions for a linear mapping T : E → F to be an extended composition operator.
This means that there exists some map ϕ : Y → βX so that T (f ) = Cϕ (f ) (: = f˜◦ ϕ),
f ∈ E. Here f˜ stands for the Stone extension of f . We also characterize those maps ϕ
for which Cϕ satisfies one of the following conditions: 1) Cϕ is continuous; 2) Cϕ maps
some 0-neighbourhood into a bounded set; 3) Cϕ maps some 0-neighbourhood into an
equicontinuous, a compact or a weakly compact set; 4) Cϕ maps any bounded set into
an equicontinuous, a compact or a weakly compact set.
1 – Introduction and preliminaries
Let X be a Hausdorff completely regular space and K the real or complex field.
Denote by F(X) (resp. B(X), C(X)) the algebra of all K-valued (resp. bounded
K-valued, continuous K-valued) functions on X and put Cb (X) : = C(X) ∩ B(X).
We will say that u ∈ F(X) vanishes at infinity if, for every ² > 0, some compact
K ⊂ X exists with |u(x)| < ² for all x ∈
/ K. The set of all f ∈ C(X) vanishing at
infinity is denoted by C0 (X). A Nachbin family on X is any collection V of non
negative upper semicontinuous (in short u.s.c.) real functions such that:
1) for every x ∈ X, there exists some v ∈ V with v(x) 6= 0, and
2) for every v1 , v2 ∈ V and λ > 0, there exists v ∈ V so that λ vi ≤ v, i = 1, 2.
Received : February 25, 1999; Revised : June 7, 1999.
AMS Subject Classification: 47B38, 46E10, 46E25.
Keywords and Phrases: Extended composition operator; (locally) Compact operator;
Strongly bounded operator; (locally) Equicontinuous operator; Weighted space.
330
L. OUBBI
The weighted spaces associated with V are:
CV (X) : =
and
CV0 (X) : =
½
f ∈ C(X) : sup v(t) |f (t)| < +∞, ∀ v ∈ V
½
t∈X
¾
f ∈ C(X) : f v vanishes at infinity for every v ∈ V
¾
.
These spaces are supplied with the weighted topology τV generated by the seminorms
f 7→ |f |v : = sup v(t) |f (t)|, v ∈ V .
t∈X
The most used Nachbin families on X are:
n
o
Z : = non negative constant functions ,
n
K : = positive multiples of characteristic functions of compact subsets of X
n
o
S0+ : = non negative u.s.c. functions on X vanishing at infinity ,
n
F : = positive multiples of characteristic functions of finite subsets of X
o
o
,
.
The corresponding weighted spaces are respectively CZ(X) = Cb (X) and
CZ0 (X) = C0 (X) with the uniform convergence topology σ, CK(X) = CK0 (X) =
C(X) with the compact open topology τc , CS0+ (X) = C(S0+ )0 (X) = Cb (X) with
the strict topology β and CF (X) = CF0 (X) = C(X) with the pointwise convergence one τs . For further details concerning the weighted spaces, see [1] and [9].
In this paper we deal with extended composition operators from arbitrary
subspaces E of a weighted space CV (X) into another weighted space CU (Y ).
Historically, the composition operators on Hilbert spaces L2 (X), X = R, Z or
[0, 1], appeared originally in the work [3] of B.O. Koopmann in connection with
classical mechanics. A survey on such operators on spaces of type L2 and H 2 was
given by E.A. Nordgren [7]. Then the composition operators were generalized in
several directions. In [2] H. Kamowitz gave a characterization of compact composition operators on the algebra C(X) for X compact. While R.K. Singh and
W.H. Summers studied in [12] the composition operators on general weighted
spaces of type CV (X) and CV0 (X). There, composition operators were characterized among the linear mappings T from CV (X) or CV0 (X) into itself. It was
also given necessary and sufficient conditions on ϕ : X → X so that Cϕ : f 7→ f◦ ϕ
maps continuously CV (X) or CV0 (X) into itself. In [4] the authors consider the
331
EXTENDED COMPOSITION OPERATORS IN WEIGHTED SPACES
special weighted space CK(X), but allow T to land in CK(Y ) for some other
Hausdorff completely regular space Y . They then gave necessary and sufficient
conditions on ϕ : Y → X under which the composition operator Cϕ : f 7→ f ◦ ϕ
is locally (weakly) compact, i.e. Cϕ maps every bounded subset of CK(X) into
a (weakly) compact subset of CK(Y ). Here, we not only consider composition
operators T : E → CU (Y ) from a general subspace E of CV (X) into a general
weighted space CU (Y ), but also we allow the map ϕ to take values outside of
X, namely in the Stone-Čech compactification βX of X. In this way we recover,
even in case E = CU (Y ), operators which fail to be composition ones in the
classical sense (see Example 2.2). We then characterize those (continuous) linear
operators which turn out to be extended composition ones. Moreover, we provide
necessary and sufficient conditions on ϕ : Y → βX so that the induced extended
composition operator Cϕ satisfies one of the following conditions:
1. Cϕ is continuous,
2. Cϕ maps some 0-neighbourhood into a bounded set,
3. Cϕ maps some 0-neighbourhood into an equicontinuous, a compact or a
weakly compact set, or
4. Cϕ maps any bounded set into an equicontinuous, a compact or a weakly
compact set.
Henceforth, unless the contrary is stated, all subspaces of CV (X) in consideration will be supplied with the topology induced by τV . For every such subspace
E and every v ∈ V , we will denote by Bv (E) the set {f ∈ E : |f |v ≤ 1} and by
coz(E) the set of all elements of X at which at least one f ∈ E does not vanish;
here E may be any subset of C(X). We will also consider the following sets:
n
F(E) : = x ∈ βX : f˜(x) 6= ∞, for every f ∈ E
n
M ∗ (E) : = x ∈ F(E) : f˜(x) 6= 0 for some f ∈ E
n
o
o
,
,
M + (E) : = x ∈ M ∗ (E) : δx , the evaluation at x,
is bounded on bounded subsets of E
o
.
If v ∈ V is given, using Lemma 9 of [8], one can extend v by upper semicontinuity
to M ∗ (E) by putting
n
o
ṽ(t) : = inf w(t); w : M ∗ (E) → R+ u.s.c. and w|X ≥ v .
332
L. OUBBI
This extension is minimal and verifies, for every f ∈ E,
|f |v = sup ṽ(t) |f˜(t)| .
t∈M ∗ (E)
Denote by Ve the collection of all ṽ, when v runs over V , and consider the set
M (E) : =
n
x ∈ M ∗ (E) : ṽ(x) 6= 0 for some v ∈ V
o
.
Whenever E is either a (complex) selfadjoint algebra and a Cb (X)-module, the
sets M ∗ (E), M + (E) and M (E) are nothing but the algebraic, the bounded and
the continuous characters (i.e. non zero multiplicative functionals) on E respectively [8]. They may differ from each other as shows the following example: let
X be the real line, C0+ (X) the set of all R+ -valued continuous functions on X
vanishing at infinity and δ the continuous function defined on X by δ(x) = 1 for
x ∈ N, δ(x) = 0 for x ≤ 1/2 or n + 1/2n ≤ x ≤ n + 1 − 1/2(n+1) for some n ∈ N
and δ piecewise linear. For every v ∈ C0+ (X), put uv : = max(v, δ) and consider the
Nachbin family U : = {λ uv : v ∈ C0+ (X), λ > 0}. We then have CU (X) = Cb (X)
algebraically and the topology τU is stronger than β but coarser then σ. It is
βX
clear that every x ∈ N defines a continuous character on CU (X) while the evalβX
uation at every point of {n − 1/2n, n ∈ N} \ R defines a discontinuous one.
Hence M (E) 6= M ∗ (E). To get a case where the three sets are all different, take
the product of the foregoing example and Cb (R) with the compact open topology.
This is a subalgebra of CV (Y ), where Y is the disjoint union of R and R, and V
the product U ×K (see [8]).
One may think that E can be considered as a topological subspace of
CU (M ∗ (E)) for some Nachbin family U on M ∗ (E), and then some of our results derive from known ones. This is not true, since otherwise every character
of E should be continuous.
In all what follows, X and Y will stand for Hausdorff completely regular
spaces, V and U Nachbin families respectively on X and Y and E a vector
subspace of C(X). For a linear map T : E → CU (Y ), set YT,E : = coz(T (E)) and
Ta : = Ra ◦ T , where Ra : CU (Y ) → CUa (YT,E ) denotes the restriction map and
Ua the relative Nachbin family induced by U on YT,E . Whenever YT,E coincides
with the whole of Y we say that T (E) is essential.
EXTENDED COMPOSITION OPERATORS IN WEIGHTED SPACES
333
2 – Extended composition operators
In this section we define the extended composition operators, give examples,
characterize the maps ϕ : Y → βX so that Cϕ : E → CU (Y ) is a continuous
extended composition operator and exhibit among the linear mappings T : E →
CU (Y ) those which are extended composition operators.
Definition 1. A linear map T from E into F(Y ) is called an extended
composition operator (in short e.c.o.) if there exists a map ϕ from Y into βX
such that, for every f ∈ E, T (f ) = f˜ ◦ ϕ. We will then write T = Cϕ .
Notice that ϕ must map Y into F(E), since f˜◦ ϕ is a (finite) function. For
such a map, let us denote by Ya , Yb and Yc respectively the sets ϕ−1 (M ∗ (E)),
ϕ−1 (M + (E)) and ϕ−1 (M (E)); and by ϕa , ϕb and ϕc the restriction of ϕ to Ya ,
Yb and Yc respectively. Actually Ya is nothing but coz(Cϕ (E)) = YCϕ ,E .
Examples 2.
1. Every composition operator (i.e. induced by a map ϕ : Y → X) is an e.c.o..
2. Consider X = Y = {0} ∪ [1, ∞) with the relative topology induced by R,
V = U = Z and E = CV0 (X) (= C0 (X) with the topology of the uniform
norm). If p ∈ βX\X is fixed, then put
T (f )(t) =
(
f (t) ,
f˜(p) ,
t ∈ [1, ∞),
t=0.
Then T is a continuous linear map from E into CU (Y ) and T (f g) =
T (f ) T (g) for every f, g ∈ E. If we set ϕ(t) = t for t 6= 0 and ϕ(0) = p,
we get that T = Cϕ and then T is an extended composition operator.
It is shown in [12] that T is not a composition operator (example 3.6).
3. Set X = Y = R, U = K and V the system of those weights of the form
max(v, sin), v ∈ S0+ . Put then T : CV (R) → CK(R) assigning to any f
the constant function with value f˜(t0 ); t0 being a cluster point in β R of
the set {2 n π + π2 , n ∈ N}. Then T is a continuous e.c.o. induced by the
constant map ϕ(t) = t0 .
4. The same example as in 3. with t0 a cluster point of the set {2 n π, n ∈ N}.
Here T is bounded but not continuous.
5. Take X = Y = R, E = C0 (E) and T : E → F(R) defined by T (f ) = f on
R+ and 0 elsewhere. Then T is an e.c.o. induced by the map ϕ : R → β R
defined by ϕ(t) = t for t > 0 and ϕ(t) = p for some p ∈ β R \ R elsewhere.
Here the range of T is not contained in C(Y ).
334
L. OUBBI
We now give a characterization of continuous composition operators between
weighted spaces. To avoid trivialities, we will assume that Ya 6= ∅ in the following
result.
Theorem 3. Let E ⊂ CV (X) be a Cb (X)-module and ϕ : Y → F(E) a map.
If Cϕ (E) ⊂ C(Y ), then Ya is open and ϕa is continuous. Moreover the following
are equivalent:
1. Cϕ maps continuously E into CU (Y ).
2. Cϕ (E) ⊂ C(Y ) and Ua ≤ Ve ◦ ϕa on Ya .
Proof: Let t0 ∈ Ya and Ω a neighbourhood of ϕ(t0 ) in βX. There is
some f ∈ E and g ∈ C(βX) with f˜(ϕ(t0 )) = 1, g(ϕ(t0 )) = 1, 0 ≤ g ≤ 1 and
supp(g) ⊂ Ω. Put h = f g, we still denote the restriction of g to X by g. Then
h ∈ E and one has: h̃(ϕ(t0 )) = f˜g(ϕ(t0 )) = 1. But h̃ ◦ ϕ is continuous. Hence
G : = {y ∈ Y : 12 < |h̃ ◦ ϕ(y)| < 32 } is open and contains t0 . But G ⊂ Ya and
ϕ(G) ⊂ Ω, hence Ya is open and ϕa continuous.
For the equivalence of 1. and 2., notice that ϕa is continuous in both cases.
1. ⇒ 2.: Fix y0 ∈ Ya . There exists some g ∈ E with g̃(ϕ(y0 )) = 1. The
continuity of Cϕ gives
|f˜◦ ϕ|u ≤ |f |v , f ∈ E .
∀ u ∈ U, ∃ v ∈ V :
In particular
u(y0 ) |f˜(ϕ(y0 ))| ≤ sup v(t) |f (t)| ,
f ∈E .
t∈X
For every n ∈ N, consider an open subset Ωn,1 of βX such that Ωn,1 ∩ M ∗ (E) =
{x ∈ M ∗ (E) : ṽ(x) < ṽ(ϕ(y0 )) + 1/n} and put Ωn : = Ωn,1 ∩ {x ∈ βX: 1 − 1/n <
|g̃(x)| < 1 + 1/n}. Then Ωn is a neighbourhood of ϕ(y0 ) in βX. Choose fn ∈
C(βX) with 0 ≤ fn ≤ 1, fn (ϕ(y0 )) = 1 and supp fn ⊂ Ωn . Since hn : = gfn belongs
to E, one has
u(y0 ) |h̃n (ϕ(y0 ))| ≤ sup v(t) |hn (t)| ,
t∈X
or
Whence
¯
¯
¯
¯
n∈N,
u(y0 ) ¯g̃n (ϕ(y0 )) fn (ϕ(y0 ))¯ ≤ sup ṽ(t) |hn (t)| ,
h
t∈Ωn
i
u(y0 ) ≤ ṽ(ϕ(y0 )) + 1/n (1 + 1/n) .
Passing to the limit, we get u(y0 ) ≤ ṽ ◦ ϕ(y0 ).
n∈N.
EXTENDED COMPOSITION OPERATORS IN WEIGHTED SPACES
335
2. ⇒ 1.: Assume that Ua ≤ Ve ◦ ϕa . Then for every u ∈ U , there is some v ∈ V
such that u ≤ ṽ ◦ ϕ on Ya . Hence, for every f ∈ E, f˜◦ ϕ is continuous and one
has
|f˜◦ ϕ|u = sup u(y) |f˜(ϕ(y))|
y∈Y
= sup u(y) |f˜(ϕ(y))|
y∈Ya
≤ sup ṽ(ϕ(y)) |f˜(ϕ(y))| ≤ |f |v .
y∈Ya
This gives at once f˜◦ ϕ ∈ CU (Y ) and the continuity of Cϕ .
Remark 4.
1. Example 5 above shows that, for a given ϕ : Y → F(E), the inclusion
Cϕ (E) ⊂ C(Y ) need not hold although Ya is open and ϕa continuous.
2. A consequence of the condition 2 of Theorem 3 is that, in order Cϕ to
be continuous, ϕ must map Ya into M (E). Hence Yc = Ya and ϕc = ϕa .
Furthermore the implication 2. ⇒ 1. holds although E fails to be a
Cb (X)-module.
It is known that if T is a continuous unital algebra morphism from CK(X)
into CK(Y ), then there is a continuous map ϕ from Y into X such that T = Cϕ
(see [5]). Actually the condition unital is not redundant for T to be induced
by a map ϕ. The multiplication by the characteristic function of a closed and
open subset of a disconnected space X provides a continuous algebra morphism
from CK(X) into itself. But such a morphism is not induced by any self map
ϕ : X → X. The following lemma yields an extended version of the result of [5].
Here we drop the continuity condition and take general subspaces of C(X).
Lemma 5. Let E be a selfadjoint subalgebra of C(X) which is also a
Cb (X)-module and T a linear map from E into C(Y ). Assume in addition that
either M ∗ (E) 6= F(E) or T (E) essential. Then T is an e.c.o. if and only if it is
an algebra morphism.
Proof: The necessity is trivial in both cases. For the sufficiency, assume first
that some x0 ∈ F(E)\M ∗ (E) exists and let y ∈ Y be given. If y ∈
/ YT,E , then set
ϕ(y) = x0 . Otherwise δy ◦ T is a character of E and by Corollary 14 of [8], there
is a (unique) x ∈ M ∗ (E) such that δy ◦ T = δx . Put then ϕ(y) = x and obtain
T (f )(y) = f˜(ϕ(y)) for every y ∈ Y and f ∈ E. Now, if T (E) is essential, then
YT,E = Y and we argue as above.
336
L. OUBBI
If none of the conditions M ∗ (E) 6= F(E) and T (E) is essential is fulfilled,
then T may fail to be an e.c.o.. The example given in Remark 4 yields such a
situation.
Despite its algebraic aspect, Lemma 5 provides, as an application, a characterization of continuous extended composition operators between some weighted
spaces. This includes in particular Theorem 3.3 and Theorem 3.4 of [12]. Moreover our proof does not utilize the approximation theory as in [12]. In the following, let us write Eb to mean E ∩ B(X).
Proposition 6. Let E ⊂ CV (X) be a vector space, T : E → CU (Y ) a linear
mapping and F ⊂ E an algebra which is either selfadjoint and a Cb (X)-module.
1. Suppose that YT,F = YT,E . If M ∗ (F ) ⊂ F(E) or T continuous and
M (F ) ⊂ F(E), then Ta is an e.c.o. if and only if T (f g) = T (f ) T (g)
for every f ∈ E, g ∈ F with f g ∈ F . If in addition M ∗ (E) 6= F(E), the
conclusion still holds for T instead of Ta .
2. If T is continuous and F dense in E, then Ta is an e.c.o. if and only if
T (f g) = T (f ) T (g) for every f, g ∈ F .
Proof: It is clear that if Ta or T is an e.c.o., then T (f g) = T (f ) T (g) even for
every f, g ∈ E. For the converse, by Lemma 5, there exists ϕ1 : YT,F → M ∗ (F )
such that T (f )(y) = f˜◦ ϕ1 (y) for every y ∈ YT,F and f ∈ F .
1. If f ∈ E and y ∈ YT,E are given, then there exists g ∈ F such that
g̃(ϕ1 (y)) = 1. Choose h ∈ C(βX) such that h(ϕ1 (y)) = 1, 0 ≤ h ≤ 1 and
supp h ⊂ {x ∈ βX : |f˜(x) − f˜(ϕ1 (y))| < 1}. The function f h g belongs to F
and since ϕ1 (y) ∈ F(E), we get T (f hg)(y) = fg
hg(ϕ1 (y)) = f˜(ϕ1 (y)) on one hand,
and T (f hg)(y) = T (f )(y) T (hg)(y) = T (f )(y) on the other hand. Since y is
arbitrary in YT,F = YT,E , Ta (f ) = f˜◦ ϕ1 . Now, if T is assumed to be continuous,
then the map ϕ1 takes its values in M (F ) and we only need that the latter be
contained in F(E). Furthermore if M ∗ (E) 6= F(E), we extend ϕ1 to the whole of
Y by putting ϕ(y) = x0 for a fixed x0 ∈ F(E)\M ∗ (E) and every y ∈
/ YT,E . This
gives T = Cϕ .
2. Since F is dense in E, M (F ) = M (E). Moreover the continuity of T yields
YT,F = YT,E . To conclude using 1., we only need to show that T (f g) = T (f ) T (g)
for every f ∈ E, g ∈ F with f g ∈ F . Let then f ∈ E, g ∈ F be so that f g ∈ F
and y ∈ YT,E . There exists a net (fi )i∈I ⊂ F converging
to f in³ E and some h ∈
n
C(βX) with h(ϕ1 (y)) = 1, 0 ≤ h ≤ 1 and supp h ⊂ x ∈ βX : max |f˜(x)−f˜(ϕ1 (y))|,
´
o
|g̃(x) − g̃(ϕ1 (y))| < 1 . The function f hg belongs to F and hg ∈ F is bounded
so that (fi hg)i∈I converges to f hg in E. Hence lim T (fi hg) = T (f hg). In partii
EXTENDED COMPOSITION OPERATORS IN WEIGHTED SPACES
337
cular lim T (fi hg)(y) = T (f hg)(y). But T (fi hg) = T (fi )T (hg) for every i ∈ I.
i
Then T (f hg)(y) = lim T (fi )(y) T (hg)(y).
i
The continuity of T again gives
T (f hg)(y) = T (f )(y) T (hg)(y) = T (f )(y) T (g)(y). Since y is arbitrary in YT,E ,
we get T (f ) T (g) = T (f g).
Remark 7.
1. If F happens to be contained in CV0 (X), then Fb is automatically dense
in F . On the other hand, if T (E) is essential, then T = Ta . In particular
Theorem 3.3 and Theorem 3.4 of [12] are obtained by taking respectively
in 2. of the proposition above E = CV0 (X) and F = CV0 (X) ∩ Cb (X) and
in 1. E = CV (X) and F = CV0 (X) ∩ Cb (X). Recall that in [12] CV0 (X)
is assumed to be essential and that M (CV0 (X)) always is contained in X.
2. An algebra morphism T : CK(X) → CK(Y ) need not be an e.c.o.. However, there is some open subset YT of Y and a continuous map ϕa : YT → X
such that Ta = Cϕa . Moreover, in the light of Theorem 3, T is continuous
if and only if for every compact K ⊂ Y, ϕa (K ∩ YT ) is relatively compact
in X.
3. If E ⊂ CV (X) is a vector space such that Eb is dense in E, then Eb is even
large in E and M + (E) = M + (Eb ). Indeed if B is a bounded set of E and
A : = {Λn (f ), f ∈ B, n ∈ N}, then A is bounded in Eb and B ⊂ A. Here for
every f ∈ B and n ∈ N, Λn (f ) denotes the bounded function defined by
f (x)
Λn (f )(x) = f (x) if |f (x)| ≤ n and Λn (f )(x) = n
otherwise. Now let
|f (x)|
x ∈ M + (E) be given. We have to show that δx does not vanish identically
on Eb . If f ∈ E is such that δx (f ) = 1 and (xi )i is a net in X converging
to x, then assuming with no loss of generality that |f (xi )| ≤ 2 for every i,
we have δx (Λ2 (f )) = lim Λ2 (f )(xi ) = lim f (xi ) = δx (f ) = 1. Hence
i
i
M + (E) ⊂ M + (Eb ). Next, assume that x ∈ M + (Eb ) and B is a bounded
set of E. Consider the set A as above. There exists some M > 0 so that
˜
|Λg
n (f )(x)| < M . We claim that |f (x)| ≤ M . If not, there will exist a
˜
neighbourhood O of x with |f (t)| > M for every t ∈ O. Hence, for n > M
and t ∈ X ∩ O, we have |Λn (f )(t)| > M . This leads to the contradiction
|Λg
n (f )(x)| ≥ M .
Now, if Cϕ has to land in a (smaller) subspace of CU (Y ) instead of the whole
space CU (Y ), the behaviour of ϕ turns out to be affected. To see this, we need
the following lemma containing analogous of Lemma 3.1 of [11] and Lemma 2,
p. 69 of [6].
338
L. OUBBI
Lemma 8. Let E ⊂ CV (X) be a vector subspace which is a Cb (X)-module.
If K is a compact subset of M ∗ (E) and C a closed one such that K ∩ C = ∅, then
1. There exists an open subset Ω of M ∗ (E) such that K ⊂ Ω and, for every
v ∈ V, ṽ is bounded on Ω.
2. There exists f ∈ E with f˜ = 1 on K and f˜ = 0 on C.
Proof: 1. Let x ∈ M ∗ (E) and fx ∈ E be so that ffx (x) = 1. Put Ωx : =
S
{|ffx | > 1/2}. By compacity, there exist x1 , ..., xn such that K ⊂ Ω : = ni=1 Ωxi .
Then Ω is the required open set.
2. We first associate with every g ∈ C(X), a positive bounded and continuous
1
function Γ(g) defined on X by Γ(g)(x) : = |g(x)| if |g(x)| ≤ 1 and Γ(g)(x) : =
|g(x)|
otherwise. Now let x ∈ M ∗ (E)\C and g ∈ E so that g̃(x) = 1 and consider
gx ∈ C(βX) with gx (x) = 1, 0 ≤ gx ≤ 1 and gx = 0 identically on C. The function
kx : = ggx belongs to E and so does hx : = kx kx Γ(kx2 ). Moreover 0 ≤ hx ≤ 1,
fx (x) = 1 and hx vanishes identically on C. Now, by compacity of K, there exist
h
S
Pm
hx1 , hx2 , ..., hxm in E such that K ⊂ m
i=1 {hxi > 1/2}. The function h : = n=1 hxn
belongs to E and satisfies h(t) > 1/2 for every t ∈ K. Now f : = 2 h Γ(2h) enjoys
the required conditions.
For a non negative function u on Y and ² > 0, set Nu : = {y ∈ Y : u(y) > 0}
and N (u, ²) : = {y ∈ Y : u(y) ≥ ²}. Consider next as in [8] the following algebras:
C` U(0) (Y ) : =
CA U(0) (Y ) : =
n
n
o
f ∈ CU(0) (Y ) : ∀ u ∈ U, ∃ u0 ∈ U with |f | u ≤ u0 ,
f ∈ CU(0) (Y ) : f is bounded on each Nu , u ∈ U
CuA U(0) (Y ) : = CU(0) (Y ) ∩ Cb (Y ) .
o
,
The proof of the following proposition is easy and then omitted:
Proposition 9. Let ϕ : Y → F(E) be a map such that Cϕ maps continuously
E into CU (Y ). Then
1. Cϕ (E) ⊂ C` U (Y ) if and only if for every f ∈ E and u ∈ U , there exists
u0 ∈ U such that |Cϕ (f )| ≤ u0 /u on Nu .
2. Cϕ (E) ⊂ CA U (Y ) if and only if for every u ∈ U , ϕ(Nu ) is E-bounding
(i.e. E ⊂ B(ϕ(Nu ))).
3. Cϕ (E) ⊂ CuA U (Y ) if and only if ϕ(Y ) is E-bounding.
EXTENDED COMPOSITION OPERATORS IN WEIGHTED SPACES
339
For the corresponding subspaces of CV0 (X), combine Proposition 9 and Proposition 10 below.
Proposition 10. Let ϕ : Y → F(E) be such that Cϕ maps E continuously
in CU (Y ). Consider the following assertions.
1. Cϕ (E) ⊂ CU0 (Y ).
2. ∀ u ∈ U, ε > 0 and K ⊂ M ∗ (E) compact, ϕ−1 (K) ∩ N (u, ε) is relatively
compact in Y .
3. ∀ u ∈ U, ε > 0 and v ∈ V such that u ≤ ṽ ◦ ϕ on Ya , ϕ−1 (K) ∩ N (u, ε)
is relatively compact whenever K is a compact subset of N (v, ε).
Then 1. ⇒ 2. ⇒ 3. If in addition E ⊂ CV0 (X), then also 3. ⇒ 1.
Proof: In order to show the implication 1. ⇒ 2., let u ∈ U, ε > 0 and K a
compact subset of M ∗ (E) be given. By Lemma 8, we may find f ∈ E such that
0 ≤ f ≤ 1 and f˜|K = 1. By assumption, the set Y1 : = {y ∈ Y : u(y) |f˜◦ ϕ(y)| ≥ ε}
is compact. Since ϕ−1 (K) ∩ N (u, ε) is contained in Y1 , it is relatively compact.
The assertion 3. derives obviously from 2. Now assume that E ⊂ CV0 (X) and
consider f ∈ E, u ∈ U and ε > 0. Set Y1 : = {y ∈ Y :: u(y) |f˜◦ ϕ(y)| ≥ ε}. If
v ∈ V is such that u ≤ ṽ ◦ ϕ on Ya , then Y1 ⊂ {y ∈ Y : ϕ(y) ∈ {|f | v ≥ ε}}.
But f ∈ CV0 (X), then K : = {|f | v ≥ ε} is compact and K ⊂ N (v, ε/M ), where
M : = sup{|f (t)|; t ∈ K}. By 3., ϕ−1 (K) ∩ N (u, ε/M ) is relatively compact. The
proof is achieved since Y1 is contained in ϕ−1 (K) ∩ N (u, ε/M ).
The following proposition gives a further algebraic characterization of extended composition operators. To show it, we need an additional notation. For
Y1 ⊂ Y , set ∆Y1 : = {δy , y ∈ Y1 } and let ∆ denote the set {δx , x ∈ M ∗ (E)}.
Proposition 11. Let E ⊂ C(X) be a Cb (X)-module and T : E → C(Y ) a
linear map. If T is an e.c.o., then T ∗ (∆Ya ) ⊂ ∆. Conversely if T ∗ (∆YT,E ) ⊂ ∆
(with F(E) 6= M ∗ (E) in case YT,E 6= Y ), then T = Cϕ for some ϕ : Y → F(E).
Proof: If T = Cϕ for some ϕ : Y → F(E), f ∈ E and y ∈ Ya , then T ∗ (δy )(f ) =
δy ◦ T (f ) = T (f )(y) = f˜(ϕ(y)) = δϕ(y) (f ). Since y ∈ Ya and f ∈ E were
arbitrary, the necessity follows. For the sufficiency, let y ∈ YT,E and x ∈ βX
satisfy T ∗ (δy ) = δx . The compacity of βX combined with the fact that E is a
Cb (X)-module shows that x is uniquely determined. Put then ϕ(y) = x. For
eventual y ∈
/ YT,E , put ϕ(y) = x0 , x0 being a (also eventual) fixed point of
F(E)\M ∗ (E). We then have T = Cϕ .
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L. OUBBI
3 – Compact extended composition operators
This section deals with the conditions on ϕ so that Cϕ becomes strongly
bounded, (locally) equicontinuous, (locally) weakly compact or (locally) compact.
Let S : A → B be a linear mapping from a locally convex space A into another B. Recall that S is said to be (weakly) compact (resp. locally (weakly)
compact) if it maps some 0-neighbourhood (resp. every bounded set) in A into a
relatively (weakly) compact subset of B. S is strongly bounded if it maps some
0-neighbourhood in A into a bounded subset of B. If B consists of continuous
functions on a topological space Z, then S is equicontinuous (resp. locally equicontinuous) if the image under S of some 0-neighbourhood (resp. of any bounded
set) in A is equicontinuous on Z.
Proposition 1. Let ϕ : Y → F(E) be a mapping such that Cϕ (E) ⊂ C(Y ).
1. If Cϕ (E) ⊂ CU (Y ), then Cϕ is strongly bounded if and only if there is
some v ∈ V such that Ua ≤ {λ ṽ ◦ ϕa : λ > 0}.
2. If ϕa = ϕ, then the following assertions are equivalent:
i. Cϕ is equicontinuous.
ii. Cϕ is locally equicontinuous.
iii. ϕ is locally constant.
Proof: 1. We have just to show the necessity. If Cϕ (Bv (E)) is bounded for
some v ∈ V , then for every u ∈ U , there is M > 0 with u(t) |f˜◦ ϕ(t)| ≤ M for
every t ∈ Y and f ∈ Bv (E). Fix y0 in Ya . If ṽ(ϕ(y0 )) = 0, consider, for every
n ∈ N, fn ∈ Bv (E) such that f˜n (ϕ(y0 )) = n and apply to fn the inequality above
to get that u(y0 ) = 0. Now if ṽ(ϕ(y0 )) 6= 0, the same inequality applied to any
fn ∈ Bv (E) such that f˜n (ϕ(y0 )) = 1/(ṽ(ϕ(y0 ))+1/n) gives u(y0 ) ≤ M ṽ ◦ϕ(y0 ),
see Lemma 9 of [8].
2. It is clear that i. ⇒ ii. In order to deduce iii. from ii., assume that ϕ is
not constant on any neighbourhood of some y0 ∈ Y . Fix a neighbourhood Ω of
ϕ(y0 ) in M ∗ (E) such that for every v ∈ V, ṽ is bounded on Ω. Then for every
neighbourhood G of y0 , there is some yG ∈ G ∩ ϕ−1 (Ω) such that ϕ(yG ) 6= ϕ(y0 ).
Take fG ∈ E with 0 ≤ fG ≤ 1, f˜G (ϕ(yG )) = 1, f˜G (ϕ(y0 )) = 0 and supp f˜G ⊂ Ω.
The set {fG , G a neighbourhood of y0 } is bounded in E and then Cϕ (B) is
equicontinuous on Y . But the equality |f˜G (ϕ(yG )) − f˜G (ϕ(y0 ))| = 1 for every
G contradicts the convergence of yG to y0 . The implication iii. ⇒ i. derives from
the fact that if ϕ is locally constant, then Cϕ (Bv (E)) is equicontinuous for every
v ∈V.
EXTENDED COMPOSITION OPERATORS IN WEIGHTED SPACES
341
Notice that an equicontinuous operator need not be even bounded on bounded
sets as shows the following example: Take X = Y = R, V = K, U = Z or K
and ϕ(t) = x0 for a given x0 ∈ β R\R and every t ∈ Y . By Proposition 1, Cϕ
is equicontinuous from E : = Cb (R) into CU (Y ). However if B : = {fn : n ∈ N},
where fn (x) = min(|x|, n) for every x ∈ R, then B is bounded in E while Cϕ (B) is
not bounded in CU (Y ). This example shows that Theorem A of [2] is extendible
neither to Schmets algebras CP (X) (even for compact X) nor to weighted spaces
in the present sense. Actually the necessary and sufficient condition for Cϕ in
Theorem A to be locally compact turns out to be here equivalent to the equicontinuity of Cϕ . In order to get the local compacity, we need some more conditions.
We will say that a subset X1 of a topological space X satisfies the sequences
cluster point property (SCP P ) if every (infinite) sequence of X1 has a cluster
point in X. Every relatively countably compact subspace of X has (SCP P ).
In particular every relatively compact and every relatively sequentially compact
subset of X has (SCP P ).
Proposition 2. Assume that ϕ(Y ) ⊂ M ∗ (E). If Cϕ : E → CU (Y ) is locally
weakly compact, then for every subset K of Y satisfying (SCPP), ϕ(K) is finite.
Proof: Assume that K ⊂ Y satisfies (SCP P ) and ϕ(K) is infinite. Let
(xn )n = (ϕ(yn ))n be a sequence of pairewise disctinct points of ϕ(K). By assumption on K, there is some y ∈ Y such that y ∈ {yn , n ≥ m} for every m.
The continuity of ϕ gives x : = ϕ(y) ∈ {xn , n ≥ m} for every m. Let G be a
neighbourhood in M ∗ (E) of x : = ϕ(y) such that, for every v ∈ V , ṽ is bounded
on G. We assume, with no loss of generality, that x 6= xn , n ∈ N. We can then
find fn ∈ E with 0 ≤ fn ≤ 1, f˜n (x) = 1, f˜n (xk ) = 0, for k ≤ n and f˜n vanishes
outside of G. Then the set B : = {fn : n ∈ N} is bounded in E and hence Cϕ (B)
is relatively weakly compact. If g is a (weakly) cluster point of Cϕ (B), then
σ(CU (Y ),CU (Y )0 )
, m ∈ N. Therefore g(t) ∈ {f˜n ◦ ϕ(t), n ≥ m}
g ∈ {f˜n ◦ ϕ, n ≥ m}
for every m ∈ N and t ∈ Y . Whereby g(y) = 1 and g(ym ) = 0 for every m. This
is in contradiction with y ∈ {yn , n ≥ m}.
In the following, if A is a subset of M ∗ (E), we will denote by EA the locally
convex space obtained by endowing E with the topology of pointwise convergence
on A. In case A = M ∗ (E), M + (E) or M (E), we will write Es∗ , Es+ or Es . We
then get
Lemma 3. Let A ⊂ F(E) and ϕ : Y → F(E) be given. If Cϕ : EA → CU (Y )
is continuous, then ϕ(Nu ∩ Ya ) is a finite subset of A for every u ∈ U . The
converse is true provided U ⊂ B(Y ) (or Cϕ : E → CU (Y ) continuous).
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L. OUBBI
Proof: For every u ∈ U , there is some finite set S ⊂ A and some M > 0
such that
u(t) |f˜◦ ϕ(t)| ≤ M sup |f˜(s)| ,
t ∈ Y and f ∈ E .
s∈S
This is only possible if ϕ(Nu ∩ Ya ) ⊂ S. Otherwise, the inequality above would
not hold for any t ∈ Nu ∩ Ya with ϕ(t) ∈
/ S and any f ∈ E such that f˜(ϕ(t)) = 1
and f˜|S = 0. For the converse if U ⊂ B(Y ) and S : = ϕ(Nu ∩ Ya ) is a finite
subset of A, then for every f ∈ E, we have |Cϕ (f )|u ≤ kuk∞ sups∈S |f˜(s)|. Now,
if Cϕ : E → CU (Y ) is continuous, for u ∈ U , there is some v ∈ V verifying
ua ≤ ṽ ◦ ϕa . If S : = ϕ(Nu ∩ Ya ) ⊂ A is finite, then
|f˜ ◦ ϕ|u =
≤
sup u(t) |f˜◦ ϕ(t)|
t∈Nu ∩Ya
sup ṽ ◦ ϕ(t) |f˜◦ ϕ(t)|
t∈Nu ∩Ya
≤ |ṽ|S |f˜|S .
Whence the conclusion.
The following result gives a sufficient condition under which Cϕ is locally
compact.
Proposition 4. Assume that ϕ(Y ) ⊂ M + (E), CU (Y ) is quasi-complete
and either U ⊂ B(Y ) or Cϕ : E → CU (Y ) continuous. If ϕ(Nu ) is finite for every
u ∈ U , then Cϕ : E → CU (Y ) is locally compact.
Proof: By Theorem 3, Cϕ : Es+ → CU (Y ) is continuous. Then Cϕ also is
continuous from (E, σ(E, E + )) into CU (Y ), where E + is the bounded dual of E.
Since E and (E, σ(E, E + )) have the same bounded sets, every bounded set B
of E is precompact in (E, σ(E, E + )). Hence for every bounded subset B of E,
Cϕ (B) is compact, for it is precompact and complete.
The condition ϕ(Y ) ⊂ M + (E) is not superfluous as shows the example after
Proposition 1. Notice also that if Nu satisfies (SCP P ) for every u ∈ U , then U is
automatically contained in B(Y ). Indeed if u ∈ U is not bounded on Y , there is a
sequence (yn )n ⊂ Y such that u(yn ) > max(n, u(yn−1 ) + 1). If y is a cluster point
of {yn : n ∈ N} and G : = {t ∈ Y : u(t) < u(y) + 1}, then G ∩ {yn , n ≥ m} 6= ∅
for every m ∈ N. This gives u(y) ≥ m for every m, which is a contradiction.
Combining Proposition 2, Proposition 4 and the foregoing remark, we get the
main result of this section:
EXTENDED COMPOSITION OPERATORS IN WEIGHTED SPACES
343
Theorem 5. Assume that ϕ(Y ) ⊂ M + (E), CU (Y ) is quasi-complete and Nu
satisfies (SCP P ) for every u ∈ U . Then the following assertions are equivalent:
1. Cϕ is locally compact from E into CU (Y ).
2. Cϕ is locally weakly compact from E into CU (Y ).
3. For every u ∈ U, ϕ(Nu ) is finite.
Next, if H ⊂ CU (Y ) and u ∈ U are given, we will say that uH vanishes
at infinity, if for every ε > 0, there exists a compact set K ⊂ Y outside of which
u|f | < ε for every f ∈ H. It is clear that for every bounded subset H of CU (Y ) and
every u ∈ U , uH vanishes at infinity whenever U fulfils the following condition:
For every u ∈ U , there exists u0 ∈ U such that for every ε > 0, there exists a
compact set K ⊂ Y with u(t) ≤ ε u0 (t) for every t ∈
/ K. We then obtain
Theorem 6. Let Cϕ : E → CU (Y ) be an e.c.o. Assume that for every
u ∈ U and every bounded subset H of Cϕ (E), uH vanishes at infinity. Then Cϕ
is compact provided it is strongly bounded and equicontinuous. The converse is
true whenever Y is a UR -space.
Proof: Let v ∈ V be such that H : = Cϕ (Bv ) is either bounded and equiconτ
tinuous. Then the pointwise closure H s of H also is either contained in CU (Y ),
τ
(τs -) bounded and equicontinuous. By Ascoli’s theorem H s is compact in the
compact open topology τc . To conclude, it is sufficient to show that τV is coarser
τ
τ
that τc on H s . Let then f0 ∈ H s , u ∈ U and ε > 0 be given. By our assumption, there exists a compact K ⊂ Y out of which u|f | < ε/2 for every
τ
f ∈ H s . Set M : = sup{u(t), t ∈ K} and ε0 : = min(ε/M, ε/2). Then the set
τ
τ
{f ∈ H s : |f (t) − f0 (t)| ≤ ε0 , t ∈ K} is a τc -neighbourhood of f0 in H s . Since
τ
it is contained in {f ∈ H s : |f − f0 |u ≤ ε}, the result follows. The converse is a
consequence of Lemma 2.3 of [10].
REFERENCES
[1] Bierstedt, K.D. – Gewichtete Raüme stetiger Funktionen und das projective
Tensorprodukt I, J. Reine Ang. Math., 259 (1973), 186–210.
[2] Kamowitz, H. – Compact weighted endomorphisms of C(X), Proc. Amer. Math.
Soc., 83(3) (1981), 517–521.
[3] Koopmann, B.O. – Hamiltonian systems and transformations in Hilbert space,
Proc. Nat. Acad. Sci. U.S.A., 17 (1931), 315–318.
344
L. OUBBI
[4] Lindstöm, M. and Lliavona, J. – Compact and weakly compact homomorphisms between algebras of continuous functions, J. Math. Anal. Appl., 166 (1992),
325–330.
[5] Llavona, J.G. and Jaramillo, J.A. – Homomorphisms between algebras of
continuous functions, Can. J. Math., 81(1) (1989), 132–162.
[6] Nachbin, L. – Elements of approximation theory, Math. Studies, 14, van Nostrand, Princeton, N.J., 1967.
[7] Nordgren, E.A. – Composition operators on Hilbert spaces, Lect. Notes in
Math., 693, Springer Verlag, Berlin (1978), 37–63.
[8] Oubbi, L. – On different algebras contained in CV (X), Bull. Belgian Math. Soc.
(Simon Stevin), 6 (1999), 111–120.
[9] Prolla, J.B. – Topological algebras of vector-valued continuous functions, Adv.
Math. Suppl. Studies, 7B (1981), 727–740.
[10] Ruess, W.M. and Summers, W.H. – Compactness in spaces of vector valued
continuous functions and assymptotic almost periodicity, Math. Nachr., 135 (1988),
7–33.
[11] Singh, R.K.; Manhas, J.S. and Singh, B. – Compact operators of composition
on some locally convex function spaces, J. Operator Theory, 31(A) (1994), 11–20.
[12] Singh, R.K. and Summers, W.H. – Composition operators on weighted spaces
of continuous functions, J. Austral. Math. Soc., 45(A) (1988), 303–319.
L. Oubbi,
Department of Mathematics, Ecole Normale Supérieure de Rabat,
B.P. 5118, Takaddoum, 10105 Rabat – MOROCCO